Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 1579468, 5 pages
http://dx.doi.org/10.1155/2016/1579468

Research Article

Sharp One-Parameter Mean Bounds for Yang Mean

Wei-Mao Qian,¹ Yu-Ming Chu,² and Xiao-Hui Zhang²

¹School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
²School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Received 19 July 2015; Accepted 9 February 2016

Academic Editor: Yann Favennec

Copyright © 2016 Wei-Mao Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We prove that the double inequality holds for all with if and only if and , where , and , , and are the Yang and th one-parameter means of and , respectively.

1. Introduction

Let and with . Then the th one-parameter mean , th power mean , harmonic mean , geometric mean , logarithmic mean , first Seiffert mean , identric mean , arithmetic mean , Yang mean , second Seiffert mean , and quadratic mean are, respectively, defined by

It is well known that both the means and are continuous and strictly increasing with respect to for fixed with . Recently, the one-parameter mean and Yang mean have attracted the attention of many researchers.

Alzer [1] proved that the inequalities

hold for all with and .

In [2, 3], the authors discussed the monotonicity and logarithmic convexity properties of the one-parameter mean .

In [4, 5], the authors proved that the double inequalities

hold for all with and if and only if , , , and .

Xia et al. [6] proved that the double inequality

holds for all with if , and inequality (4) is reversed if .

Gao and Niu [7] presented the best possible parameters and such that the double inequality holds for all with and .

In [8, 9], the authors proved that the double inequalities

hold for all with if and only if , , , and .

Xia et al. [10] found that is the best possible lower power mean bound for the one-parameter mean if and is the best possible upper power mean bound for the one-parameter mean if .

For all with , Yang [11] provided the bounds for the Yang mean in terms of other bivariate means as follows:

In [12, 13], the authors proved that the double inequalities

hold for all with if and only if , , , , , and , where is the unique solution of the equation on the interval , and .

Very recently, Zhou et al. [14] proved that and are the best possible parameters such that the double inequality

holds for all with .

The aim of this paper is to present the best possible parameters and such that the double inequality holds for all with .

2. Main Result

In order to prove our main result we need a lemma, which we present in this section.

Lemma 1. Let , and

Then the following statements are true:

(1)if , then for all ;(2)if , then there exists such that for and for .

Proof. For part (1), if , then (9) becomes

Therefore, part (1) follows from (10).
For part (2), let , , , , , , , , , , and . Then elaborated computations lead to

Note that

It follows from (24) and (25) that

for .
From (23) and (26) we clearly see that is strictly increasing with respect to on the interval . Then (21) and (22) lead to the conclusion that there exists such that the function is strictly decreasing on and strictly increasing on .
It follows from (19) and (20) together with the piecewise monotonicity of the function that there exists such that the function is strictly decreasing on and strictly increasing on .
Making use of (13)–(18) and the same method as the above we know that there exists such that the function is strictly decreasing on and strictly increasing on .
It follows from (12) and the piecewise monotonicity of the function that there exists such that the function is strictly decreasing on and strictly increasing on .
Therefore, part (2) follows easily from (11) and the piecewise monotonicity of the function .

Theorem 2. The double inequality

holds for all with if and only if and .

Proof. Since and are symmetric and homogeneous of degree one, without loss of generality, we assume that and . Let and . Then (1) lead to

where

where is defined by (9).
We divide the proof into four cases.
Case 1 ( ). Then it follows from Lemma 1(2), (29), and (31) that there exists such that the function is strictly decreasing on and strictly increasing on , and

Therefore,

follows easily from (28), (30), and (32) together with the piecewise monotonicity of the function .
Case 2 ( ). Then (1) leads to

Inequality (34) implies that there exists large enough such that for all with .
Case 3 ( ). Then from Lemma 1(1) and (31) we know that the function is strictly increasing on the interval . Therefore,

follows from (28) and (30) together with the monotonicity of the function .
Case 4 ( ). Let and ; then making use of Taylor expansion we get

Equation (36) implies that there exists small enough such that for all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401191 and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-15G17.

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